Universal Hirschberg for Width Bounded Dynamic Programs

TL;DR

This paper generalizes Hirschberg's space-efficient algorithm to a broad class of width-bounded dynamic programs on DAGs, enabling deterministic traceback with significantly reduced space in various sequence analysis problems.

Contribution

It introduces a general framework for space-efficient traceback in width-bounded dynamic programs on DAGs, extending Hirschberg's idea beyond sequence alignment.

Findings

Deterministic traceback in space $O( ext{width} imes ext{log} T)$ for width-bounded DPs.

Near-optimal space bounds for sequence alignment and graph-based DPs.

Structural limitations on space efficiency in streaming models.

Abstract

Hirschberg's algorithm (1975) reduces the space complexity for the longest common subsequence problem from $O(N^2)$ to $O(N)$ via recursive midpoint bisection on a grid dynamic program (DP). We show that the underlying idea generalizes to a broad class of dynamic programs with local dependencies on directed acyclic graphs (DP DAGs). Modeling a DP as deterministic time evolution over a topologically ordered DAG with frontier width $\omega$ and bounded in-degree, and assuming a max-type semiring with deterministic tie breaking, we prove that in a standard offline random-access model any such DP admits deterministic traceback in space $O(\omega \log T + (\log T)^{O(1)})$ cells over a fixed finite alphabet, where $T$ is the number of states. Our construction replaces backward dynamic programs by forward-only recomputation and organizes the time order into a height-compressed recursion tree whose nodes expose small "middle frontiers'' across which every optimal path must pass. The framework yields near-optimal traceback bounds for asymmetric and banded sequence alignment, one-dimensional recurrences, and dynamic-programming formulations on graphs of bounded pathwidth. We also show that an $\Omega(\omega)$ space term (in bits) is unavoidable in forward single-pass models and discuss conjectured $\sqrt{T}$-type barriers in streaming settings, supporting the view that space-efficient traceback is a structural property of width-bounded DP DAGs rather than a peculiarity of grid-based algorithms.